2       I   ELEMENTS OF THE THEORY OF SETS
and jc are considered as " variables" (and sometimes it is x which is fixed,
and/which becomes the "varying" object).
Section 1.9 gives the most elementary properties of denumerable sets;
this is the beginning of the vast theory of " cardinal numbers" developed by
Cantor and his followers, and for which the interested reader may consult
Bourbaki ([3], Chapter III) or (for more details) Bachmann [2]. It turns out,
however, that, with the exception of the negative result that the real numbers
do not form a denumerable set (see (2.2.17)), one very seldom needs more
than these elementary properties in the applications of set theory to analysis.
1. ELEMENTS AND SETS
We are dealing with objects, some of which are called sets. Objects are
susceptible of having properties, or relations with one another. Objects are
denoted by symbols (chiefly letters), properties or relations by combinations
of the symbols of the objects which are involved in them, and of some other
symbols, characteristic of the property or relation under consideration.
The relation x = y means that the objects denoted by the symbols x and y
are the same; its negation is written x =£ y.
If X is a set, the relation x e X means that x is an element of the set X,
or belongs to X; the negation of that relation is written x $ X.
If X and Y are two sets, the relation X c Y means that every element
of X is an element of Y (in other words, it is equivalent to the relation
(Vx)(x e X => x e Y)); we have X c X, and the relation (X <= Y and Y c Z)
implies X c Z. If X c Y and Y c X, then X = Y, in other words, two
sets are equal if and only if they have the same elements. If X c Y, one says
that X is contained in Y, or that Y contains X, or that X is a subset of Y; one
also writes Y z> X. The negation of X c Y is written X dp Y.
Given a set X, and a property P, there is a unique subset of X whose
elements are all elements x e X for which P(x) is true; that subset is written
{xeX\P(x)}. The relation {x e X|P(x)} c {x eX|Q(x)} is equivalent to
(V* eX)(P(x) => q(jc)) ; the relation {x e X | P(;c)} = {x e X | Q(x)} is equivalent
to (VjeeX)(P(jc)oQ(x)). We have, for instance, X = {xeX\x = x} and
X = {x e XI x e X}. The set 0X = {x e X \ x ^ x} is called the empty set of X;
it contains no element. If £ is any property, the relation x e 0X => P(x) is
true for every x, since the negation of x e 0X is true for every x (remember
that Q=>P means "not Q or P"). Therefore, if X and Y are sets, x e 0X
implies x e 0V, in other words 0X c 0Y, and similarly 0Y c= 0X, hence
0x = 0v» all empty sets are equal, hence noted 0.
If a is an object, the set having a as unique element is written {a}.